Description of the numerical testbed use for the trombone in the continuous and discrete time. Transfer functions A, B, C, D, T and R are given in equations X, from which transfer functions $\hat{A}$, $\hat{B}$, $\hat{C}$ and $\hat{D}$, $\hat{T}$ and $\hat{R}$ are deduced using the bilinear transform: $\forall z\in ℂ\mathrm{s}.\mathrm{t}.|z|<1$, $s=2f_s\frac{1-z^{-1}}{1+z^{-1}}$, and given in equations (23a)–(23f), using $\kappa =\frac{r_0}{2\pi {\nu }_c({\theta }_0)c_0}$.